A beam finite element for the analysis of structures in fire

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by Richard Shaun Walls

Dissertation presented for the Degree of Doctor of Philosophy

in the Faculty of Engineering, at Stellenbosch University

Supervisor: Dr Celeste Viljoen

Co-supervisor: Dr Hennie de Clercq

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Declaration

By submitting this dissertation electronically, I declare that the entirety of the work contained therein is my own original work, that I am the authorship owner thereof (unless to the extent explicitly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any qualification.

Signature:

Date:

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Abstract

All building structures require a specified fire resistance rating and numerous procedures have been produced for ensuring this. In engineering practice designers can generally not perform detailed structural fire designs on buildings due to the high computational modelling requirements of most modern structures, and so they typically resort to conservative prescriptive methods instead. Hence, design engineer orientated methods are required to improve fire safety while providing more economical buildings. The goal of this dissertation is to provide a simple, but technically accurate, model for the analysis of structures in fire, including composite structures, which considers buildings as skeletal frames.

To achieve this end a beam finite element has been developed that has a moving, eccentric neutral axis that accounts for material properties that change as structures heat up. A composite bending stiffness, axial stiffness and resultant thermal forces are calculated for a generic cross-section. Material and geometric nonlinearity is considered. The properties of any number of materials (e.g. a steel beam, concrete slab and reinforcing steel) are represented by single beam properties. These calculated beam properties can be included in either commercially available, but simple, finite element software or advanced finite element modelling tools. The only assumption required is that Euler-Bernoulli behaviour, where plane sections remain plane, must hold. A methodology for including rebar tension stiffening at elevated temperatures has been included based on modifying an ambient temperature model.

A series of numerical case studies are presented, comparing the results of the proposed beam formulation against finite element models using shell elements. Results between these models (which includes deflections, stresses, strains and neutral axis positions) typically differ by 0-5% when Euler-Bernoulli assumptions hold. Furthermore, case studies and experimental results from real fire tests in the literature were also analysed by the proposed formulation coupled with relatively simple finite element software. The deflections of structures in fire predicted by the proposed model are well within acceptable tolerances for fire engineering systems, and typically comparable to more complex models in the literature. The model developed has been used to investigate eleven different beams consisting of steel beams, concrete slabs and composite steel-concrete beams, along with conducting a series of parametric studies. With further research and the inclusion of three-dimensional behaviour the method could become a valuable tool for the analysis of structures in fire.

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Opsomming

Alle geboustrukture vereis 'n bepaalde brandbestandheid gradering en talle prosedures bestaan om dit te verseker. Ontwerpers in ingenieurspraktyk is in die algemeen nie in staat om gedetailleerde rasionale brand ontwerpe vir strukture uit te voer nie, as gevolg van die hoë numeriese modellering vereistes vir meeste moderne strukture. Daarom gebruik ingenieurs tipies konserwatiewe voorskriftelike metodes. Dus is eenvoudiger, ontwerp-georiënteerde modellering metodes nodig om brandveiligheid te verbeter terwyl meer ekonomiese geboue verskaf word. Die doel van hierdie verhandeling is om 'n eenvoudiger, maar tegnies akkuraat, model vir die analiese van strukture, insluitend saamgestelde strukture, in brande te voorsien waarin geboue as skeletale rame beskou word.

Om hierdie doel to bereik is ‘n balk eindige element ontwikkel wat ‘n bewegende, eksentriese neutrale as (NA) gebruik om die veranderinge in die eienskappe van materiale in ag te neem. ‘n Saamgestelde buigingstyfheid, aksialestyfheid en resulterende temperatuurkragte word vir ‘n generiese dwarssnit bereken. Materiaal en geometriese nielineariteit is beskou. Die eienskappe van ‘n aantal materiale (bv. ‘n staalbalk, betonblad en bewapeningstaal) word deur enkele balk eienskappe verteenwoordig. Hierdie berekende balk eienskappe kan ingesluit word in kommersiël beskikbare, maar eenvoudige, eindige element sagteware, of gevorderde eindige element modellering gereedskap. Die enigste benodigde aanname is dat Euler-Bernoulli gedrag, waar ‘n gegewe dwarsnit in ‘n enkele vlak bly, moet gebruik word in die analiese. 'n Metode vir die insluiting van bewapeningstaal trekspanning verstywing by hoë temperature is ingesluit, wat ontwikkel is deur 'n wysiging van 'n kamertemperatuur model.

‘n Aantal numeriese gevallestudies word aangebied. Resultate van die metode en ‘n Abaqus model met gebruik van dop elemente is vergelyk. Die resultate van die modelle (wat defleksies, spanning, vervorming en NA posisies insluit) is tipies binne 0-5% van mekaar wanneer Euler-Bernoulli aannames gebruik word. Verder is gevallestudies en eksperimentele resultate uit die literatuur ook geanalieseer met gebruik van die metode gekoppel met relatief eenvoudige eindige element sagteware. Die defleksies vir brand situasies soos bereken met gebruik van die voorgestelde model is binne aanvaarbare toleransies vir brand ingenieurswese stelsels, en is tipies vergelykbaar met meer komplekse modelle uit die literatuur. Die voorgestelde model is gebruik om elf verskillende balke, wat bestaan uit staal balke, betonblaaie en saamgestelde balke, te ondersoek. 'n Reeks van parametriese studies is ook uitgevoer. Met verdere navorsing en die insluiting van driedimensionele gedrag kan die metode 'n waardevolle hulpmiddel word vir die analiese van strukture in 'n brande.

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Acknowledgements

I would like to thank the following people for the significant contributions made towards this work:

 My supervisors, Dr Celeste Viljoen and Dr Hennie de Clercq, for their help, assistance and unending patience during the process of this research. Their guidance made a tremendous impact on this dissertation. Also, without Hennie’s crazy idea of getting fire engineering going in South Africa who knows where we would be now.

 Prof Johan Retief for his mentorship and guidance kindly provided over the past years, even before I joined Stellenbosch. He was very instrumental in setting up this PhD, as well as providing insight and assistance throughout.

 My colleagues in the Department of Civil Engineering.

 To Prof Charles Clifton for the discussions, documents and feedback over the past few years.

 To my parents for their support and encouragement over many years of study, and providing the opportunities that they did.

 To my wife, Merryn, for her unending love, patience and help over the past years, even when things got tough. Without her support I would not have been able to do this. And also to our first child who is on his/her way, making sure that there was a definite deadline for the submission of this dissertation.

“For God so loved the world that he gave his one and only Son,

that whoever believes in him shall not perish but have eternal life.” – Jesus (John 3:16)

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List of figures

Figure 1.1: Design and analysis models available for structural fire engineering showing the position of the

FBE formulation (adapted from Stadler (2012) with FE model picture from Clifton (2014)) ... 3

Figure 1.2: Structure adopted for dissertation to address overall objectives ... 7

Figure 2.1: Time-temperature curves of the standard, external and hydrocarbon fires ... 12

Figure 2.2: Time-temperature behaviour of a real fire ... 12

Figure 2.3: Testing and modelling regimes in fire engineering with their associated level of credibility according to Gales et al (2012). Letters in brackets cite works listed in the original paper. ... 14

Figure 2.4: Market share in the UK of various fire protection systems (Tata Steel & BCSA 2013) ... 15

Figure 2.5: Failed column and beams at the Cardington fire tests (Lamont 2001) ... 19

Figure 2.6: Change in load carrying behaviour for a composite floor exposed to increasing temperatures (Bailey 2002) ... 20

Figure 2.7: Deflected shape of a composite structure considered by Clifton (2014) in the development of the Slab Panel Method (SPM) ... 21

Figure 2.8: Potential failure mechanisms in slab panels with various boundary conditions (Abu & Burgess, 2010) ... 21

Figure 2.9: Moment-rotation behaviour of a variety of steel connections in fire (Al-Jabri 1999) ... 23

Figure 2.10: Behaviour of an end plate connections at 450°C – experimental and numerical results (Anderson 2011) ... 23

Figure 2.11: Finite element configurations commonly used in the nonlinear analysis of composite slabs (Stadler 2012) ... 25

Figure 2.12: In (a) a typical simply-supported slab is shown in which tensile membrane action can easily develop. However, in (b) offset columns and irregular bay sizes are experienced which may cause problems in the development of tensile membrane action (Flint et al. 2013). ... 27

Figure 3.1: Behaviour of a uni-axially loaded beam subjected to mechanical and thermal loads ... 34

Figure 3.2: Analysis Step (a) – Rectangular beam with mechanical load applied ... 37

Figure 3.3: Analysis Step (b) – Uniform increase in temperature of a portion of the beam, with no shear strength assumed for the section ... 37

Figure 3.4: Analysis Step (c) - Upward curvature of beam due to cross-sectional restraint ... 37

Figure 3.5: Analysis Step (d) – Converting temperature effects into resultant RTSL/M forces ... 38

Figure 3.6: Analysis Step (e) – Calculation of deflections based on applied forces ... 38

Figure 3.7: Analysis Step (f) – Determination of mechanical strains from total strains ... 39

Figure 3.8: Analysis Step (g) – Calculation of mechanical stresses based on mechanical strains ... 39

Figure 3.9: Stresses in a rectangular cantilever with the lower two-fifths uniformly heated ... 40

Figure 3.10: Analysis model for a cantilever with a uniformly heated lower portion ... 41

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Figure 3.12: Changes in neutral axis and bending stiffness of an IPE 200 due to the Young's modulus varying

... 43

Figure 3.13: Relationship between the upward thermal curvature / bowing and steel temperature of a composite beam with variable slab thickness (Bailey 1995). Note how the scenario with h = 0mm (no slab) has the maximum upward deflection when in reality it should have zero deflection. ... 44

Figure 3.14: Cantilever beam with the lower portion uniformly heated, an end point load and with material properties dependent on strain and temperature ... 45

Figure 3.15: Design procedure for using the eccentric beam formulation ... 46

Figure 3.16: Temperature, thermal strain, Young's modulus and ETS profiles over the height of a composite section used for determining Nθ and Mθ values. ... 48

Figure 3.17: Flowchart illustrating the iterative procedure used for determining the non-linear properties of cross-sections exposed to fire. ... 49

Figure 3.18: Flowchart for the nonlinear analysis of structures subjected to mechanical and thermal loads (the latter can be neglected as required) (based on Iu & Chan (2004)) ... 50

Figure 3.19: Typical layout and details considered for the design of a composite structure. ... 51

Figure 3.20: Analysis and design of slab panels independent of primary beams. Yield line patterns are used to determine the loading for subsequent steps. ... 52

Figure 3.21: Analysis model of the composite system ... 53

Figure 3.22: Typical temperature profile in a composite slab exposed to fire ... 53

Figure 3.23: Analysis model showing eccentric neutral axis positions, calculated bending moments and general structural behaviour ... 55

Figure 3.24: Modelling methodology and steps for analysing beams in fire using commercial FE software ... 56

Figure 4.1: Variation in Young's modulus and strain across a composite section, showing section discretisation to account for such behaviour ... 63

Figure 4.2: Flow diagram for the determination of the updated neutral axis and stiffness values of a beam cross-section ... 65

Figure 4.3: Layout showing how a beam is modelled along a reference axis but has an updated neutral axis about which the beam deforms ... 66

Figure 4.4: Relationship between reference axis and deformed configuration ... 66

Figure 4.5: (a) Plane element before deformations, but noting global degrees of freedom (XY axis). (b) The element after deformation and motion showing deformations in the local (x’y’) axis (based on Cook et al (2001)) ... 69

Figure 4.6: Stress-strain relationships for the theoretical material model used ... 71

Figure 4.7: Case Study A - Simple cantilever ... 72

Figure 4.8: Analysis behaviour and considerations for Case Study A ... 73

Figure 4.9: Vertical deflections of Case Study A with increasing end moment with E0 = 200 GPa ... 74

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Figure 4.11: Vertical deflection of the simple cantilever with end moment, shear and axial forces applied .... 75

Figure 4.12: Case Study B - 5x300 rectangular beam with a UDL showing the Abaqus and final FBE model 76 Figure 4.13: Stresses over cross-sections at various positions for Case Study B ... 76

Figure 4.14: Deflection with increasing load for Case Study B ... 77

Figure 4.15: Isoline strain plot for Case Study B. The central green line of zero strain represents the NA. ... 78

Figure 4.16: Case Study C – Fixed-fixed IPE 200 showing structural layout and final FBE configuration ... 79

Figure 4.17: Deflected shape and maximum stresses in the Abaqus model of Case Study C ... 79

Figure 4.18: Stress profiles for Case Study C at various location ... 80

Figure 4.19: Rectangular beam with UDL. The Young’s modulus is varied to create stiffer zones internally. 81 Figure 4.20: Half of the Abaqus model of Case Study D showing longitudinal strains in the section ... 82

Figure 4.21: Cross-sectional strain profile for Case Study D at different locations ... 82

Figure 4.22: Deflection of Case Study D with increasing load ... 83

Figure 5.1: Steel thermal elongation as a function of temperature ... 87

Figure 5.2: Specific heat of steel as a function of temperature ... 88

Figure 5.3: Thermal conductivity as a function of temperature ... 88

Figure 5.4: Reduction factors for various steel properties at elevated temperatures ... 89

Figure 5.5: Stress-strain curve for steelwork at elevated temperature including strain-hardening (ECCS 2001) ... 90

Figure 5.6: Equivalent thermal stress to cause thermal elongation in unrestrained steelwork ... 91

Figure 5.7: Non-uniform variation in steel temperature over the length of beams in structures (Franssen & Vila Real 2010) ... 92

Figure 5.8: Stress-strain curves according to EN 2-1-1 (BSI 2004) accounting to the confinement of concrete, comparing confined and unconfined concrete samples ... 94

Figure 5.9: Stress-strain model of concrete at elevated temperatures according to EN 2-1-2 (BSI 2005a) ... 95

Figure 5.10: Comparison of stress-strain and Esecant-strain for EN 2-1-1 and EN 2-1-2 for fcm = 30MPa at ambient temperature ... 96

Figure 5.11: Thermal elongation of siliceous and calcareous concrete to EN 2-1-2 (BSI 2005a) ... 97

Figure 5.12: Specific heat of concrete as a function of temperature according to EN 2-1-2 (BSI 2005a), with moisture contents, u, of 0%, 1.5% and 3% ... 98

Figure 5.13: Upper and lower limits of thermal conductivity according to EN 2-1-2 (BSI 2005a) ... 99

Figure 5.14: Comparison of temperature profiles in a 100mm concrete slab after a 30, 60 and 120 minute fire according to (a) EN 2-1-2, (b) EN 4-1-2, (c) Wickström, and (d) a FEA model developed in this research. . 101

Figure 5.15: Sheeting profiles for ribbed composite slabs and their equivalent thicknesses ... 102

Figure 5.16: Temperature profiles in a profiled slab, illustrating (a) the actual profiles, (b) the simplified profile adopted in this research and by Stadler (2012), and (c) an alternative formulation that could be used in this research for 2D temperature profiles ... 103

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Figure 5.18: Tension stiffening model for concrete developed by Deeny (2010) ... 106

Figure 5.19: Load-strain relationship of a reinforced concrete sample under tensile loading (fib 2010a) ... 107

Figure 5.20: Effective concrete areas of reinforced elements in tension: (a) beams, (b) slabs, and (c) member in tension (fib 2010a)... 109

Figure 5.21: Load-strain relationship of a reinforced concrete sample under tensile loading, including the modified graph for cracking load exceeding the rebar yield load ... 110

Figure 5.22: Load-strain graph to illustrate changing properties with increasing temperature ... 111

Figure 5.23: Change in the tension stiffening area for rebar due to the complex strain profiles of concrete slabs in fire. ... 112

Figure 5.24: Graph of axial load to axial stiffness and rebar strain of a 100x100 concrete section in which both tension stiffening (T.S.) and no tension stiffening (No T.S.) are considered ... 116

Figure 6.1: Case study of a simply-supported steel beam subject to a UDL with varying levels of discretisation ... 121

Figure 6.2: Slab layout for tests conducted by Ali et al (2008) ... 122

Figure 6.3: Deflection results of the test, Ali et al's numerical model (2008), and the proposed FBE model 123 Figure 6.4: Position of the NA from the slab soffit and magnitude of the bending stiffness (EIθ) along the length of the slab at 60 minutes. ... 124

Figure 6.5: Experimental setup for Test 15 and 16 (reproduced from Wainman & Kirby (1988)) ... 125

Figure 6.6: Temperatures of elements for Test 15 ... 126

Figure 6.7: Temperatures of elements for Test 16 ... 126

Figure 6.8: Finite element modelling philosophy employed by OpenSEES (Jiang et al. 2014) ... 127

Figure 6.9: Deflection vs. time results for Test 15 showing experimental results, the FBE results and results from Jiang et al (2014) ... 128

Figure 6.10: Deflection vs. time results for Test 16 showing experimental results, the FBE results and results from Jiang et al (2014) ... 129

Figure 6.11: Second Munich test experimental layout ... 130

Figure 6.12: Finite element model developed by Stadler (2012) for the Second Munich Test ... 131

Figure 6.13: Temperatures in the Second Munich Test for steel beams (Mensinger et al. 2012; Stadler 2012) ... 133

Figure 6.14: Concrete slab temperature profile at 40 minutes... 133

Figure 6.15: Loading on beams for the Second Munich test showing yield line patterns considered... 135

Figure 6.16: Comparison of deflections between experimental data (EXPERIM.), predictions by Stadler (2012) and the FBE formulation (FBE). All readings are vertical deflections in mm. Additional results from the FBE model are provided considering top & bottom beams as continuous (BM CONT) and the slab as discontinuous (SLAB DIS) ... 137

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Figure 7.2: Change in bending stiffness of the IPE 240 with different temperatures and temperature gradients. The average temperature of each beam is the same at each specific web temperature. ... 143 Figure 7.3: Change in bending stiffness and RTSM for the IPE 240 with varying temperature gradients and web at 400°C ... 144 Figure 7.4: Graph of bending stiffness against mechanical axial load for the IPE 240 beam. For the graph compression is negative. ... 145 Figure 7.5: Concrete slab used for the parametric study with reinforcement placed on the bottom ... 146 Figure 7.6: Stress profile in a 200x500 deep reinforced concrete beam at failure ... 147 Figure 7.7: Graph showing temperature over the height of the 130mm slab when exposed to different periods of a standard fire ... 148 Figure 7.8: Graph showing the total, mechanical and thermal strains over the height of the 130mm slab for a 60 minute standard fire exposure time ... 149 Figure 7.9: Mechanical strains over the height of the 130mm slab for different standard fire exposure times. ... 149 Figure 7.10: Graph showing secant Young's modulus values over the height of the slab at different fire exposure times ... 150 Figure 7.11: Graph showing the internal stresses caused by mechanical strains over the height of the 130mm slab when exposed to different periods of a standard fire. ... 150 Figure 7.12: Graph of bending stiffness and RTSM against standard fire time for the concrete slab with no applied mechanical forces, with (a) neither tension capacity (fctm = 0 MPa) nor tension stiffening (TS), (b) for

fctm = 2.6 MPa but no TS, and (c) for both fctm = 2.6 MPa and TS. ... 151

Figure 7.13: Deflections of a 3m long cantilever slab with varying tensile properties when heated in a standard fire ... 152 Figure 7.14: Graph of bending stiffness and Mθ against applied mechanical load for the 130mm concrete slab

... 153 Figure 7.15: Graph of axial stiffness and thermal axial force against applied mechanical load for the concrete slab... 154 Figure 7.16: Graph of bending stiffness and Mθ against concrete strength ... 154

Figure 7.17: Graph of bending stiffness and Mθ against standard fire time exposure for different temperature

profiles in a 100m thick slab ... 156 Figure 7.18: Change in bending stiffness with applied axial compressive load for the concrete slab. For the graph compression is negative. ... 157 Figure 7.19: Change in RTSL, Nθ, with increasing applied axial load for the concrete slab ... 158

Figure 7.20: Cross-section considered for the parametric study based on a typical office block beam size ... 159 Figure 7.21: Temperature of the top flange and bottom flange / web for the IPE 240 when exposed to a standard fire. Bare steel (Bare) and a 10mm layer of perlite passive protection (Protected) are considered. . 159

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Figure 7.22: Graph showing the total (𝛆total), mechanical (𝛆𝛔) and thermal strains (εθ) over the height of the

composite beam when exposed to a 30 minute fire and fctm = 0 MPa ... 160

Figure 7.23: Graph showing the total (𝛆total), mechanical (𝛆𝛔) and thermal strains (εθ) over the height of the

composite beam when exposed to a 30 minute fire, a 45 kNm mechanical moment is applied and fctm = 0 MPa

... 161 Figure 7.24: Graph showing the change in mechanical stresses, σσ, in the composite beam for mechanical

moments of 0 kNm and 45 kNm respectively, with fctm = 0 MPa ... 161

Figure 7.25: Graph showing the total, mechanical and thermal strains over the height of the composite beam ... 162 Figure 7.26: Graph showing mechanical stress over the height of the composite beam ... 162 Figure 7.27: Graph of bending stiffness against applied mechanical moment for the composite beam with the passively protected steel beam after a 60 minute fire, with (a) neither tension capacity (fctm = 0 MPa) nor

tension stiffening (TS) (b) for fctm = 2.6 MPa but no TS, and (c) with both fctm = 2.6 MPa and TS. ... 163

Figure 7.28: Graph of Mθ against applied mechanical moment for the composite beam with the passively

protected steel beam after a 60 minute fire ... 163 Figure 7.29: Graph of deflection against applied mechanical moment for the composite beam after a 60 minute fire when considered as a 3m long cantilever with an end point moment ... 164 Figure 7.30: Graph of EIθ against standard fire time for the passively protected composite beam... 164

Figure 7.31: Graph of Mθ against standard fire time for the passively protected composite beam ... 165

Figure 7.32: Graph of Mθ against standard fire time for the composite slab with varying mechanical moment

and tensile properties ... 166 Figure 7.33: Graph of bending stiffness (EIθ) and RTSM (Mθ) against width of concrete flange acting

compositely with the steel beam for various values of applied mechanical moment (M) ... 167 Figure 7.34: Deflection of a 3m long cantilever subjected to an end point moment of 0kNm, 61.5kNm, and 123kNm with varying width of concrete flange acting compositely ... 168 Figure 7.35: Graph of bending stiffness and Mθ against applied mechanical load for the composite slab with

applied mechanical moments of 0.0kNm, 61.5kNm and 123kNm. For the graph compression is negative. .. 168 Figure 10.1: Temperature data for the rebar of Case Study 2 ... 185

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List of tables

Table 2.1: Cost of steel and passive protection for various column sizes in South Africa ... 10

Table 5.1: Concrete properties according to EN 2-1-1 (BSI 2004) ... 94

Table 5.2: Parameters for stress-strain profiles for siliceous concrete in fire according to EN 2-1-2 (BSI 2005c) ... 96

Table 6.1: Temperatures of elements for Test 15 ... 125

Table 6.2: Temperatures of elements for Test 16 ... 126

Table 6.3: Steel temperatures for the Second Munich Test ... 134

Table 7.1: Comparison of yield load against RTSL for different temperatures of the unloaded IPE 240 ... 146

Table 7.2: Summary of thermal stiffness, RTSL and axial restraint load for the 130mm slab exposed to different standard fire times. ... 158

Table 10.1: Case Study A - IPE cantilever – Properties for segments at different moments ... 181

Table 10.2: Case Study B – 5x300 Beam - Segment properties for a UDL of 10 kN/m ... 182

Table 10.3: Case Study C - Fixed-fixed IPE 200 - Segment properties for a UDL of 10 kN/m ... 182

Table 10.4: Case Study D - Fixed-fixed 5x300 beam - Segment properties for a UDL of 10 kN/m ... 183

Table 10.5: Case Study 1 - Segment properties for the 4 segment beam at 300°C ... 183

Table 10.11: Case Study 2 – 1200x200 concrete slab - Segment properties at 30 minutes ... 185

Table 10.12: Case Study 2 - 1200x200 concrete slab - Segment properties at 60 minutes ... 185

Table 10.13: Case Study 3 – Test 15 – Segment properties at 15 minutes ... 186

Table 10.19: Case Study 4 – Munich 2 – Edge left - Segment properties ... 188

Table 10.20: Case Study 4 – Munich 2 – Intermediate - Segment properties ... 188

Table 10.21: Case Study 4 – Munich 2 – Right left - Segment properties ... 188

Table 10.22: Case Study 4 – Munich 2 – Edge top left - Segment properties ... 189

Table 10.23: Case Study 4 – Munich 2 – Edge top right - Segment properties ... 189

Table 10.24: Case Study 4 – Munich 2 – Edge bottom left - Segment properties ... 189

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List of abbreviations

ASFP Association for Specialist Fire Protection

BS British Standard

BSI British Standards Institution

CEC Commission of the European Communities CSA Canadian Standards Association

DOF Degree of Freedom

EN European Norm (Eurocode document) ETS Equivalent Thermal Stress

EU European Union

FBE Fire Beam Element

FE Finite Element

fib Fédération internationale du béton FEA Finite Element Analysis

ISO International Standards Organisation

NA Neutral Axis

NIST National Institute for Science and Technology NFPA National Fire Protection Association

RTSL Resultant Thermal Strain Load RTSM Resultant Thermal Strain Moment SABS South African Bureau of Standards SANS South African National Standard SFPE Society for Fire Protection Engineers

SPM Slab Panel Method

TS Tension stiffening

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List of symbols

Roman

𝑎 Parameter 1 for calculation of EN 3-1-2 stress-strain curve of steelwork

𝐴 Cross-sectional area

𝑏 Breadth

𝑏 Parameter 2 for calculation of EN 3-1-2 stress-strain curve of steelwork 𝑏₀ Transverse spacing of shear studs

𝑏_𝑐𝑖 Effective width of concrete on each side of a composite beam 𝑏_𝑒𝑓𝑓 Effective concrete flange breadth

𝑏_𝑖 Geometric width of a composite slab 𝑐 Distance of neutral axis from reference axis

𝑐 Parameter 3 for calculation of EN 3-1-2 stress-strain curve of steelwork 𝑐0 Initial distance of neutral axis from reference axis

𝑐_𝑎 Specific heat of steelwork 𝑐_𝑝 Specific heat of concrete

𝐶_𝑦𝜃 Steel yield load at temperature 𝜃 𝑑 Effective depth of a concrete element

𝐸 Young’s modulus

𝐸0 Initial tangent Young’s modulus 𝐸𝑐𝑚 Mean Young’s modulus of concrete

𝐸_𝑆 Secant Young’s modulus

𝐸_𝑇 Tangent Young’s modulus

𝐸_𝜃 Young’s modulus at temperature 𝜃

𝐸𝐴 Axial stiffness

𝐸𝐴𝜃 Axial stiffness at temperature 𝜃

𝐸𝐼 Bending stiffness

𝐸𝐼_𝑆 Secant bending stiffness 𝐸𝐼_𝑇 Tangent bending stiffness

𝐸𝐼_𝜃 Secant bending stiffness at temperature 𝜃

𝑓_𝑐 Stress in concrete

𝑓_𝑐𝑘 Characteristic cylinder strength of concrete in compression 𝑓𝑐𝑘,𝑐𝑢𝑏𝑒 Characteristic cube strength of concrete in compression 𝑓𝑐𝑚 Mean cylinder strength of concrete in compression 𝑓𝑐𝑡𝑚 Mean cylinder strength of concrete in tension 𝒇_𝒊 Load vector on node i

𝑓_𝑦 Yield stress of steelwork

𝑓_𝑢 Ultimate strength of steelwork

𝑭 Load vector

𝑭_𝒎 Force vector of mechanically applied loads 𝑭𝑹 Resultant force vector including reactions ℎ Height or thickness of an element

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𝐺 Shear modulus

𝐼 Second moment of area

𝐼_{𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒} Second moment of area of a composite beam based on elastic properties

𝐼_𝑒𝑓𝑓 Effective second moment of area of a composite beam considering shear connectors 𝐼_{𝑠𝑡𝑒𝑒𝑙} Second moment of area of a steel beam

𝒌 Local stiffness matrix

𝑘_∗,𝜃 Reduction factor of material property * at temperature 𝜃

𝑲 Stiffness matrix

𝐾𝑓 Shear force shape factor

𝑙 Length of beam

𝐽 Polar moment of area

𝑀 Bending moment

𝑀_𝜃 Resultant Thermal Strain Moment (RTSM)

n Number of elements

𝑁 Axial load

𝑁_𝑟 Tension force at which cracking occurs in a reinforced concrete element 𝑁_𝑦 Yield load of a reinforced element

𝑁_𝜃 Resultant Thermal Strain Load (RTSL)

𝑵𝜽 Vector of Resultant Thermal Strain Loads (RTSL) 𝑝 Percentage of shear connection

𝑃 Applied point load

𝑄 Shear force

𝑸 Matrix relating initial nodal coordinate to updated coordinate system 𝒓 Local unbalanced restoring force vector

𝑹_𝑹 Global unbalanced restoring force vector

𝑡 Time

𝑡 Slab thickness

𝑇 Torque

𝑻 Transformation matrix relating local and global axis systems

𝑢_1𝑥 Nodal displacement (subscripts 1/2 denote start and end node, x/y denote axis) 𝑢 Moisture content of concrete

𝑈 Internal work of a system

𝑊 External work of a system

𝑾 Transpose matrix relating forces in updated to initial coordinate system 𝑥 Distance of the neutral axis from the top of a reinforced concrete element 𝑥_𝑖 Horizontal nodal displacement of node i

𝑦 Distance from axis

𝑦_𝑖 Vertical nodal displacement of node i 𝑍_𝑝𝑙 Plastic section modulus

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xxiii Greek

𝛼_𝑒 Modular ratio

𝛼_𝑖 Rotation of node i

𝛼𝜃 Coefficient of thermal expansion at temperature 𝜃 𝛽 Slope of 𝐸𝑇− 𝜎 for the non-linear material model

𝛿 Nodal deflection

∆ Nodal deflection vector

∆ ∗ Change in property * ∆_{𝑡𝑜𝑡𝑎𝑙} Total deflection of structure

∆_{𝑚𝑒𝑐ℎ} Deflection due to mechanical loading ∆𝜃 Deflection due to thermal loading

𝜀 Total strain

𝜀𝑐1,𝜃 Strain of concrete at maximum stress 𝜀𝑐𝑟𝑎𝑐𝑘𝑖𝑛𝑔,𝜃 Cracking strain of concrete at temperature 𝜃 𝜀𝑐𝑟𝑒𝑒𝑝 Creep strain

𝜀_{𝑐𝑢1,𝜃} Strain of concrete at failure 𝜀_𝑡𝑟 Transient strain

ε_θ Thermal strain

𝜀_𝜎 Mechanical strain

𝜆_𝑎 Thermal conductivity of steelwork 𝜆_𝑐 Thermal conductivity of concrete 𝜃_𝑎 Temperature of steelwork 𝜃𝑐 Temperature of concrete

𝜃𝑔 Gas temperature

𝜌 Material density

𝜌_{𝑠,𝑒𝑓𝑓} Effective reinforcement ratio for tension steel

𝜎 Stress

𝜎_𝜎 Mechanical stress

𝜎_𝜃 Equivalent Thermal Stress

𝜑 Nodal rotation

∅ Reinforcement bar diameter

Subscripts

a Steelwork

bot. fl. Bottom flange

c Concrete

el Element

E Young’s modulus

max Maximum

p Proportional limit for steelwork

top fl. Top flange

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xxiv web Web 𝑥 Major axis 𝑦 Minor axis 𝑦 Yield strength 𝜃 Temperature

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1 Chapter 1: Introduction

1.1 Background to study

Events such as the collapse of the World Trade Centre have increased the interest and rate of research in structural fire engineering worldwide in recent years. A report from the Federal Emergency Management Agency (2002) following this disaster stated that: “The behaviour of the structural system under fire conditions should be considered as an integral part of structural design.” The structural engineering industry is slowly moving towards rational structural fire design as a core issue of engineering responsibility rather than an addendum addressed after the main design work is complete. All buildings require some level of fire resistance, although structural engineers often lack the skills and knowledge required to correctly provide this.

It is commonly acknowledged that fires are complex events which involve high levels of uncertainty (Buchanan 2001). The structural response associated with such unpredictable circumstances is equally difficult to quantify. Hence, it is often debateable what level of accuracy is possible in structural fire designs. Furthermore, most developmental work and standards are based on a semi-arbitrary benchmark, the Standard Fire (ISO 1999), which has limited resemblance to real fires. Thus, it is not possible to claim a high level of precision when modelling structures in fire because of such variability regarding input factors and fire scenarios.

During the course of this research and especially during a trip to the United Kingdom in 2015 the author corresponded with a number of international universities and consulting companies involved in structural fire engineering. The general attitude expressed was that full structural fire analyses are rarely done in industry due to the extensive modelling and analysis times required for such designs. Generally, simplified models or prescriptive methods are utilised by consultants instead. This can lead to overly conservative and less economical solutions, although non-conservative designs are also possible. A leading fire consulting company noted that it was only for the upmarket buildings in central London that full analyses were typically done since the higher consulting fees on such projects justified the hours required to complete analyses, whereas other projects simply did not.

Structural steelwork, as a material in isolation, is particularly sensitive to elevated temperatures. Other construction materials, such as concrete, have a greater thermal capacity and are good insulators, while some materials, such as wood, can char which provides a protective coating. This may be an important reason why many engineers use concrete for multi-storey buildings in South Africa rather than steelwork, as the passive protection (e.g. intumescent paints) required for steelwork can be prohibitively expensive. However, research has shown that significant fire resistance can be obtained from steel structures using correct design procedures (Clifton 2013; Wang et al. 2012), without such structures becoming uneconomical.

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From the discussions above it can be seen that there is a divide between the extensive fire engineering research and analysis tools that are available and the current needs of practitioners. By focussing on producing tools suitable for design offices rather than only for research structural fire design can make a greater impact on society by assisting practitioners in safely producing more economical buildings. Thus, there is a need to develop technically accurate analysis tools which do not require extensive modelling times to produce predictions. With the highly unpredictable nature of fire it is justifiable that there be a trade-off between high levels of precision and pragmatic considerations, otherwise tools will become neglected except for rare cases.

1.2 Overview

This dissertation presents a generalised beam finite element (FE) and analysis methodology for the analysis of structural frames subject to severe fires. The aim is to provide a simpler, but technically accurate, modelling tool that could be adopted within the practical structural engineering design environment. Rather than requiring shell or volume elements only beam elements are used. By providing tools that are more accessible to engineering practitioners the design of buildings in fire can be more readily carried out, thereby providing a safer and more economical built environment. The proposed formulation considers nonlinear behaviour, temperature effects, thermal curvatures and global structural interactions, which existing simplified methods generally do not. Conversely, it allows for the modelling of structures as skeletal frames, including composite structures, which simplifies the modelling process significantly.

In order to provide a convenient ‘handle’ for the approach described in this dissertation it is called the Fire Beam Element (FBE) formulation, to differentiate it from the numerous other analysis, material and thermal models discussed. The finite element formulation presented does not only apply to structures in fire, but potentially also to other types of structures such as bridge decks or wind towers. The proposed FBE is an adaptation of the well-known Euler beam. While the Euler beam typically assumes that the bending stiffness (𝐸𝐼), axial stiffness (𝐸𝐴) and neutral axis (NA) position remain unchanged during analysis the FBE requires (1) the calculation of the position of the NA, (2) which is used to calculate updated section stiffnesses to account for the influence of (3) generalised temperature profiles, internal forces and nonlinear material properties. Large deflections typically must be accounted for due to the deformations that occur during fires. The FBE can be included in models with varying levels of complexity ranging from (a) simplified 2D analyses using commercial software to (b) advanced models where restraint, time-dependent properties and non-linear structural interactions are explicitly considered (although additional research is required for such implementation). The procedures developed are suitable for any structural configuration provided that (i) the temperature-stress-strain behaviour of constituent materials is known, and (ii) Euler-Bernoulli assumptions of plane sections remaining plane hold. When the element is included with the analysis methodologies and techniques developed in this dissertation it is referred to as the FBE formulation.

The positioning of the research presented in this dissertation is compared to existing design methods in Figure 1.1. It can be seen that there are a variety of options available for designing structures in fire, ranging from

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very simplified prescriptive methods with no consideration of overall building behaviour to advanced models where all structural elements are explicitly accounted for. The proposed prediction model can consider global building behaviour and structural interactions without requiring extensive modelling of floors and the inclusion of finite elements such as shells, rigid links or the explicit modelling of reinforcing steel. Hence, the model generally falls between the advanced and simplified analysis procedures. The design of composite floors, as opposed to composite beams, is not directly considered, and it must be combined with one of the existing tensile membrane models (Bailey & Moore 2000b; Clifton & Abu 2014; Wu et al. 2012) to consider all structural components. Details regarding the advanced methods listed in Figure 1.1 are contained in Section 2.6.

Figure 1.1: Design and analysis models available for structural fire engineering showing the position of the FBE formulation (adapted from Stadler (2012) with FE model picture from Clifton (2014))

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The FBE formulation proposed in this work allows parametric investigations to be more easily carried out, enabling designers to identify parameters which have a significant influence on results. Engineering judgement can then be used to address important parameters and make decisions regarding design criteria. Furthermore, the FBE modelling procedures could allow multiple configurations to be tested in significantly less time than would be required for full three-dimensional models consisting of shell and volume elements. This would enable engineers to evaluate multiple structural configurations and determine which system would be most suitable, especially for complicated structures.

1.3 Research objectives

The ultimate goal of the research is to produce a simplified but technically accurate method for the analysis of structures in fire. To this end the research objectives of this dissertation can be broadly defined as:

a) To examine, understand and explain the fundamental structural mechanics of a beam cross-section exposed to fire. Based on this understanding thermal effects can be converted into usable resultant thermal forces.

b) The development of a mathematical finite element formulation for beam elements in fire where (a) the position of the neutral axis (NA) can change, and (b) nonlinear material behaviour occurs.

c) To carefully consider various material and thermal input parameters for FBE models as provided in the literature and to identify those which are most suitable for predicting structural behaviour. A tension stiffening model will be incorporated, based on modifying an ambient temperature model. This is included to account for the concrete-rebar interaction that occurs, and to determine to what extent it influences calculated deflections and stresses.

d) Implementation of the FBE procedures in a computer program to validate the methodologies associated with updating the NA and the stiffness of a member. This will be done independent of thermal considerations such that calculated strain profiles and stiffnesses can be more easily identified relative to the formulations developed.

e) To investigate the influence of input parameters on FBE models through a parametric study, such that it may be ascertained to what extent parameters may influence validation case studies investigated. This study is also important for identifying structural behaviour that occurs due to cross-sectional interactions and thermal effects within a beam.

f) Implementation of the FBE formulation within simple, commercially available structural analysis software to illustrate how simplified analyses can be carried out by design engineers.

g) Validation of the FBE formulation against full-scale experimental tests. This important aspect illustrates how full systems can be considered in a manner that is efficient in terms of design engineer input requirements but still captures the behaviour of a structural system.

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1.4 Scope of the work

The FBE formulation is presented for the analysis of generic cross-sections and structural configurations. This is done such that the methodology can be applied in a wider range of scenarios, rather than only to specific types of structures, e.g. composite buildings. However, bending about only a single axis has been considered (i.e. 2D analyses), although proposals are provided for the extension of the method to be used in 3D structures. The analysis philosophy will remain the same for the 2D and 3D systems, but the latter requires additional research regarding torsion and bending behaviour about the minor axis of floor sections (as discussed in Section 8.4). Material models have been included based on existing standards and literature. Experimental work is required for the validation of the modified tension stiffening model. Plane sections must remain plane for the methodologies presented to hold. However, it is investigated in Chapter 4 to what extent this assumption influences results if it is violated.

A nonlinear analysis programme has been developed in this research for considering the numerical validation studies conducted in Chapter 4. No thermal behaviour has been considered for these studies investigated. In Chapter 6 the FBE formulation has been used in conjunction with a commercially available software package such that it can be illustrated how designs can be carried out in a simple manner. However, additional developmental work is required to fully implement the FBE formulation in software where both iterative procedures and thermal procedures are simultaneously included.

For the parametric studies conducted in Chapter 7 only individual beam cross-sections have been considered, rather than entire structures. Parametric studies of full 3D structures can be addressed in future work. Boundary conditions in all chapters have been modelled as either being fixed or pinned, as commonly done in the literature. It would be possible to include non-linear springs to model connections, and this is a topic for future research.

1.5 Outline of dissertation

The objectives above are addressed in the following manner: Chapter 2 provides an overview of literature on structural fire engineering, important concepts in design, behaviour of structures in fire and other such factors.

In Chapter 3 the procedure to be followed when analysing a full structure as a skeletal frame with beam elements is presented. It is shown how stresses and strains develop in sections exposed to thermal and mechanical effects. A procedure for determining resultant thermal forces that cause the same strains as thermal effects is proposed. This is followed by the fundamental structural mechanics and design procedures used for the FBE formulation.

In Chapter 4 the finite element (FE) methodology for updating the position of the neutral axis and the calculation of resultant beam stiffnesses is developed, which is followed by a numerical validation process. This is carried out independently of temperature considerations, by developing a generic methodology.

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Material models and temperature considerations required for fire are then presented in Chapter 5. A procedure for considering tension stiffening of reinforcing steel is also developed in this chapter. Various temperature profiles and material models are discussed.

Thereafter case studies consisting of numerical and full-scale experiments in the literature are investigated in Chapter 6 to validate the FBE formulation. It is shown that results from the proposed prediction model are comparable to more advanced methods in the literature, and the FBE formulation is able to predict experimentally observed deflections.

A parametric study is carried out in Chapter 7 which considers how specific input parameters influence calculated results. This highlights which parameters an analysis is most sensitive to, along with identifying how structures respond to various conditions. The change in bending stiffness, axial stiffness and resultant thermal forces for typical sections are plotted with varying input parameters. This chapter illustrates how the predicted deflections calculated in Chapter 6 may change based on the input parameters assumed.

Conclusions and recommendations for future research are presented in Chapter 8. Additional data can be found in Appendix A and B, which consist of analysis data used for case studies and temperature profiles developed for slabs. This developmental process is illustrated in Figure 1.2, where the most important content of each chapter is listed.

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2 Chapter 2: Literature Review

2.1 Introduction

This chapter provides the necessary background required to give context to the remaining chapters. It also provides an overview of the literature and state of knowledge related to structural fire engineering. Initially the fields of fire engineering and structural fire design are explained, along with important concepts related to fire design. The behaviour of structures in fire is then addressed, on the basis of fundamental structural mechanics. Well-known full-scale fire tests and case studies are presented, illustrating the current state-of-the-art of structural fire engineering and what behaviour has been observed from real structures in fire. Thereafter different design approaches and methodologies relating to fire design are discussed, addressing both prescriptive and performance based design. A brief discussion regarding finite element formulations is presented as a basis for the FBE formulation developed in Chapter 4. Software and design systems available for analysis are then addressed. It is shown how structures in fire are currently modelled, which is necessary for understanding how the research discussed in this dissertation contrasts with those systems already available. Due to the extensive treatment given to material models in fire presented in Chapter 5 the literature on material behaviour is not covered in this chapter.

2.2 Fire engineering and the role of structural fire design

Structural fire engineering is a broad discipline encompassing a number of specialist fields such as structural design, thermodynamics, analysis of non-linear systems, consideration of building codes, personnel safety requirements, and much more. This literature review is limited to aspects most relevant to this research topic. Refer to publications such as those of the SFPE (2008) for an in-depth discussion of the aforementioned topics.

2.2.1 Fire and society

Fire has played a significant role in the development of societies, both positive and negative. Throughout the centuries large conflagrations have occurred across the world, often destroying thousands of homes and even ruining cities and communities. For a fascinating work on historical fire events see the work by Bankoff et al (2012). In recent times the influence of fire in developed countries has been drastically reduced due to aspects such as better building code requirements and enforcement, fire resistant construction, firefighting facilities, access to water, societal education, electrification of homes, fire breaks between buildings and other such factors. However, ensuring that structures have adequate performance in the unlikely event of a fire is still an important task to be addressed by building designers.

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2.2.2 Objectives of structural fire engineering

Ultimately the aim of fire safety in buildings is to reduce to acceptable levels the probability of death or injury of persons, loss of property and damage to the environment (Bailey 2004b). The Commission of the European Communities (CEC) outlines the general requirements of construction works subjected to fire conditions as:

 “the load bearing capacity of the construction can be assumed for a specific period of time,  the generation and spread of fire and smoke within the works are limited,

 the spread of fire to neighbouring construction work is limited,  occupants can leave the works or be rescued by other means,  the safety of rescue teams is taken into consideration.” (CEC 1988)

It is often questioned whether the significant expenses incurred to provide buildings that are safe in fire are fully justified. The following are interesting facts regarding fires in Europe, as presented by Twilt (1994): (a) The likelihood of a person being killed in a car accident is 30 times higher than being killed in a building fire. (b) In a survey of 5 European countries between 74% (Netherlands) and 85% (France) of fatal fires occurred in domestic buildings. Hence, deaths in commercial and industrial structures are rare, yet this is where structural fire analyses are used. (c) The cause of deaths in buildings due to heat and smoke is generally between 74% (Germany) and 99% (Switzerland). Thus, it must be understood that almost no deaths are caused by building collapse. (d) A survey showed that the monetary loss due to fires is in the order of about 0.2-0.29% as a portion of Gross National Product (which amounts to billions of euros or dollars in a large economy). However, of the cost of damages to buildings and businesses due to fires only between 21-32% relates to the fabric of the building whereas the rest is due to stock and indirect losses (e.g. productivity). Hence, it is often business continuity and insurance requirements that govern fire design, rather than purely safety.

Following on from such facts it must be acknowledged that often the task of structural fire engineering professionals is not simply to ensure life safety, but also to reduce the cost of infrastructure while retaining a suitable level of fire safety. With steel being particularly vulnerable to fire it is often passively protected with various products. The cost of protecting various 1m long column sizes in South Africa is shown in Table 2.1. Costing was obtained from a local supplier. It can be seen that for a two hour rating on a UC 203x203x46 column the cost of intumescent paint to protect the steelwork is 60% more than the cost of the steelwork itself. This highlights the need for safe but efficient solutions.

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Fire Protection Costing

60min Fire Rating 120min Fire Rating Section Mass (kg/m): Ap/V (m-1): Steel Cost (R/m): Intumescent Paint – Nullifire S707-60 Vermiculite Spray Intumescent Paint – Nullifire S707-120 Vermiculite Spray UC 152x152x23 23.3 304 R 652 R772 R418 Not possible R761 UC 203x203x46 46.2 205 R 1,294 R503 R559 R2,076 R1,019 UC 305x305x137 137 106 R 3,836 R473 R855 R1,779 R1,558

Table 2.1: Cost of steel and passive protection for various column sizes in South Africa

2.2.3 Fire design codes

To obtain a safe but economical design engineers make use of the different fire design codes available. In the USA extensive documentation on fire protection has been produced by the NFPA (National Fire Protection Association). Such guidelines are often adhered to outside America too, especially on petrochemical projects. In Europe the Eurocodes (EN documents) have a number of sections specifically dealing with structural fire design. The EN documents are typically viewed as the most technically advanced design standards in terms of fire in the world, and have been extensively drawn upon in this work. The main documents that are considered in this research are: (a) Eurocode 1-1-2: Actions on structures – Actions on structures exposed to fire (BSI 2002b); (b) Eurocode 2-1-2: Design of concrete structures – Structural fire design (BSI 2005a); (c) Eurocode 3-1-2: Design of steel structures – Structural fire design (BSI 2005b); and (d) Eurocode 4-1-2: Design of composite steel and concrete structures – Structural fire design (BSI 2005c). For a discussion regarding the compatibility between EN and South African steel codes refer to Walls & Viljoen (2016).

2.2.4 Limit state design

Fire scenarios are considered ‘accidental loading’ situations in terms of load factors selected (SABS 2011a). Accidental loads are those which are “not expected during the design life”, but when they do occur then structures should “not be damaged to an extent disproportionate to the original cause of the abnormal event” (Retief & Dunaiski 2009). Thus, in the case of a fire it should be expected that there will be damage, and for very severe fires structural components or the whole building may need to be replaced. Allowing some level of damage is typically more economically viable than trying to make all buildings immune to the effects of fire. At the fire limit state imposed loads are reduced because the probability of design levels of imposed loading occurring simultaneously with the highly improbable event of a fire is negligible. Imposed loads are taken at between 30% to 80% of their characteristic value depending on the building occupancy and design code used (BSI 2002b; CISC 2010).

According to British (BSI 2009), South African (SABS 2011c) and European (CEC 1988) codes structural elements are tested based on the criteria of (a) load-bearing capacity, (b) integrity (preventing smoke and hot gas flow between compartments) and (c) insulation (ASFP 2014). Columns only need to meet load-bearing